100 Prof. A. E. Richardson on 



The singularities of -^- are given by 



B = tanhaw, and cosh 2 uw= ±a. 



These limit the depth of the stream, and different cases 

 arise according to the relative magnitude of the roots. 

 The stream-line i/r = is a free surface over which 



V — ~~ n~ i B — tanh olwK\ 



Fig. 2 has been drawn for the special case B = 2, 0L — \ y 

 with fi = l, i. e g = i. 



The extreme cases are given by a = l, when the free 

 surface will be vertical at (/> = and will tend to curl over 



Fig. 2. 



Numbers on stream-bed 



refer to Velocity. V// fyfr>/ *"""''-85 .. ''■' Deptf " 



yvvdmzMkmmmmmw- 



and break ; and B = l when the fluid is at rest at <£ = oo on 

 the free surface. If B<1 a rigid boundary is necessary 

 over part of the stream -line yfr — Q if this particular form of 

 free surface is possible. 



Referring to fig. 2, it is interesting to note how the 

 velocity changes over the stream-bed, and how quickly a 

 nearly uniform regime is established down stream *". The 

 tendency for a more or less quiet pool to form at the bottom 

 of the drop is apparent, as is also the tendency to erosion at 

 the upper edge of the fall, and a short distance up-stream. 



Evidently a form of G(ic) containing several factors of 

 this type would give rise to cases of flow over more irregular 

 shaped stream-beds. 



(<?) Flow over a corrugated stream-bed. 



This problem has been solved approximately by Lord Kelvin 

 ('< Mathematical and Physical Papers,' vol. iv.) on the as- 

 sumption that the irregularities are small compared with the 

 depth of the stream. 



* Searle's assumption is justified. Phil. Mag*. May 1912. 





